1) Cars between 3800 and p will be sold with an expected quality (3800+p)/2
Buyers, therefore are willing to pay p=(3800+p)/2 + 200. The equilibrium price p is 4200
For p=4200, only (4200-3800)/(5800-3800)=20% of the cars will be sold
2) a) A risk neutral consumer will pay x.1400+(1-x).800 = 600x+800
b) If all holidays are mediocre, x=0 and p=800. Since the cost of booking a mediocre holiday is 1000, there would be no transactions
c) If all holidays are excellent, then x=1 and p=1400. Since the cost is 1100, then transactions could happen. In fact, there would still be an active market if
600x+800=1100 => x=1/2. However, since each individual agent has no incentive to keep selling quality holidays if they can realise and extra 100 profit, the market will collapse, x will tend to 0 and no transaction will take place
d) If it costs the same to book a quality or mediocre holiday, then it’s all a matter of reputation to the industry. The minimum number of quality holidays that would have to be booked to keep the market going is:
600x+800>1000 => x>1/3
The surplus can be calculated as:
S = 600x + 800 – 1000 = 600X – 200
In reality, the market conditions will push the travel agencies to only book quality holidays so they can maximise the surplus.
e) For scenario c, since there are no transactions as each individual is incentivised to sell mediocre holidays, banning this low quality product would generate a market with surplus 300
For scenario d, there is no need to regulatory intervention as the surplus itself will incentivise the market to only sell quality holidays
f) Travel agencies could create a self-regulation mechanism whereby the sale of mediocre holidays would be banned. Alternatively, travel agencies could signal themselves as only sellers of excellent holidays through advertisement and money back guarantees.
3) The valuation by McClean as a function of the valuation by McDirty (v):
Indeed 1.5v – 1.5 > v => v > 3
Offering p has a (p-3) probability of being accepted and the expected value E[v] is (3+p)/2, so the expected gain is:
E[G] = (p-3) ((1.5 E[v] – 1.5) – p) = (p-3) (1.5(3+p)/2 – 1.5 – p) =
(p-3)(2.25+0.75p-1.5-p) = (p-3)(-0.25p+0.75) = -0.25p^2 + 0.75p + 0.75p – 2.25
E[G] = -0.25p^2 + 1.5p – 2.25
So, for p=3.5 => E[G] = -0.0625
The optimum bid strategy would be to maximise E[G]:
E’[G] = -0.5p + 1.5 = 0 => p=3
Therefore, although the company is worth more to McClean than to McDirty, the uncertainty of the fair value of the company causes the transaction to fail.
4) a) Length of job * Higher salary – cost education less intelligent < length job * lower salary
4*50,000 – 40,000*t < 4*20,000
t > 3
b) With no signalling needed:
Uintelligent = 200,000
Uless = 80,000
If treated as a pool:
Mean w = 0.8 (50,000) + 0.2 (20,000) = 44,000
Uintelligent = 166,000
Uless = 166,000
Finally, with signalling and t=3
Uintelligent = 30,000
Uless = 80,000
For the intelligent ones, even being treated as a pool is better than having to provide signalling.
5)
a) max P(e) = R(e) – w(e) = me – (we + k) s.t. participation constraint: w(e) – c(e) > u0
For the principal w(e) – c(e) = u0 => we + k – e2/2 = 0 => we + k = e2/2
Therefore max P(e) = me – e^2/2
P’(e*) = m – e* = 0 , so e*=m
Compatibility constraint: U(e*) > U(e) for any e <> e*
So, we want max U(e)=we + k – e2/2 for e=e*
U’(e*) = 0, so w – e*
0 therefore w
e* = m.
Final constraint: U(e*) = u0
We* + k – e*2/2 = u0
m2 + k – m2/2 = 0 => k = -m2/2
Therefore w(e) = me – m2/2
b) Franchise model. Employee wants to maximise R(e) – C(e) – F and F is such that R(e*) – C(e*) – F = u0
max U(e) = me – e2/2 – F
U’(e*) = 0
m – e*= 0 => e*=m
U(e*) = U(m) = R(m) – C(m) – F = m^2 – m^2/2 – F = u0
F = m2/2
c) if u0 changes from 0 to 2, k will change so that m^2+k-m^2/2=2 => k=2-m^2/2
and F also changes so that F=m^2/2 – 2
6) a) E(R) for e=2 = 1/3 (10-w) + 2/3 (30-w) = 70/3 – w
E(R) for e=1 = 2/3 (10-w) + 1/3 (30-w) = 50/3 – w
b) For the agent to work hard, with observable effort, U(wh,2) > U(wl,1)
wh^(1/2) – (2-1) > wl^(1/2) – (1-1) => wh^(1/2) > wl^(1/2) + 1
The participation constraints are:
U(wh,2)= u0 = 1
wh^(1/2) – (2-1) = 1 => wh=4
U(wl,1)= u0 = 1
Wl^(1/2) – (1-1) = 1 => wl=1
So, for the agent to work hard, wh>4 and wl=1 and, for the agent to make low effort wh=4 and wl>1
The net revenue for the first case will be
70/3 – 4 = 58/3
and for the second case
50/3 – 1 = 47/3
d) Principal’s objective function
max E[R] = 1/3 (10 – wl) + 2/3 (30 – wh)
max 70/3 – wl/3 – 2wh/3
max 70/3 – 1/3(wl+2wh)
is equivalent to
min (wl+2wh) which, in the feasible area, gives wl=0 and wh=9
So, EU agent = 1/3 U(0,2) + 2/3 U(9,2) = 1
E[w] = 1/3. 0 + 2/3.9 = 6
On case b, wage = 4 and expected utility is the same. The wage must be higher to compensate for the additional risk the agent is taking.
e) Expected revenue for e=2 if
1/3 (10-0) + 2/3 (30-9) = 10/3 + 42/3 = 52/3
The expected revenue for e=1 was calculated in item b as 47/3, so motivating is worthwhile.
Chapter 4 - Asymetric Information
1) Cars between 3800 and p will be sold with an expected quality (3800+p)/2Buyers, therefore are willing to pay p=(3800+p)/2 + 200. The equilibrium price p is 4200
For p=4200, only (4200-3800)/(5800-3800)=20% of the cars will be sold
2) a) A risk neutral consumer will pay x.1400+(1-x).800 = 600x+800
b) If all holidays are mediocre, x=0 and p=800. Since the cost of booking a mediocre holiday is 1000, there would be no transactions
c) If all holidays are excellent, then x=1 and p=1400. Since the cost is 1100, then transactions could happen. In fact, there would still be an active market if
600x+800=1100 => x=1/2. However, since each individual agent has no incentive to keep selling quality holidays if they can realise and extra 100 profit, the market will collapse, x will tend to 0 and no transaction will take place
d) If it costs the same to book a quality or mediocre holiday, then it’s all a matter of reputation to the industry. The minimum number of quality holidays that would have to be booked to keep the market going is:
600x+800>1000 => x>1/3
The surplus can be calculated as:
S = 600x + 800 – 1000 = 600X – 200
In reality, the market conditions will push the travel agencies to only book quality holidays so they can maximise the surplus.
e) For scenario c, since there are no transactions as each individual is incentivised to sell mediocre holidays, banning this low quality product would generate a market with surplus 300
For scenario d, there is no need to regulatory intervention as the surplus itself will incentivise the market to only sell quality holidays
f) Travel agencies could create a self-regulation mechanism whereby the sale of mediocre holidays would be banned. Alternatively, travel agencies could signal themselves as only sellers of excellent holidays through advertisement and money back guarantees.
3) The valuation by McClean as a function of the valuation by McDirty (v):
Indeed 1.5v – 1.5 > v => v > 3
Offering p has a (p-3) probability of being accepted and the expected value E[v] is (3+p)/2, so the expected gain is:
E[G] = (p-3) ((1.5 E[v] – 1.5) – p) = (p-3) (1.5(3+p)/2 – 1.5 – p) =
(p-3)(2.25+0.75p-1.5-p) = (p-3)(-0.25p+0.75) = -0.25p^2 + 0.75p + 0.75p – 2.25
E[G] = -0.25p^2 + 1.5p – 2.25
So, for p=3.5 => E[G] = -0.0625
The optimum bid strategy would be to maximise E[G]:
E’[G] = -0.5p + 1.5 = 0 => p=3
Therefore, although the company is worth more to McClean than to McDirty, the uncertainty of the fair value of the company causes the transaction to fail.
4) a) Length of job * Higher salary – cost education less intelligent < length job * lower salary
4*50,000 – 40,000*t < 4*20,000
t > 3
b) With no signalling needed:
Uintelligent = 200,000
Uless = 80,000
If treated as a pool:
Mean w = 0.8 (50,000) + 0.2 (20,000) = 44,000
Uintelligent = 166,000
Uless = 166,000
Finally, with signalling and t=3
Uintelligent = 30,000
Uless = 80,000
For the intelligent ones, even being treated as a pool is better than having to provide signalling.
5)
a) max P(e) = R(e) – w(e) = me – (we + k) s.t. participation constraint: w(e) – c(e) > u0
For the principal w(e) – c(e) = u0 => we + k – e2/2 = 0 => we + k = e2/2
Therefore max P(e) = me – e^2/2
P’(e*) = m – e* = 0 , so e*=m
Compatibility constraint: U(e*) > U(e) for any e <> e*
So, we want max U(e)=we + k – e2/2 for e=e*
U’(e*) = 0, so w – e*
0 therefore w
e* = m.Final constraint: U(e*) = u0
We* + k – e*2/2 = u0
m2 + k – m2/2 = 0 => k = -m2/2
Therefore w(e) = me – m2/2
b) Franchise model. Employee wants to maximise R(e) – C(e) – F and F is such that R(e*) – C(e*) – F = u0
max U(e) = me – e2/2 – F
U’(e*) = 0
m – e*= 0 => e*=m
U(e*) = U(m) = R(m) – C(m) – F = m^2 – m^2/2 – F = u0
F = m2/2
c) if u0 changes from 0 to 2, k will change so that m^2+k-m^2/2=2 => k=2-m^2/2
and F also changes so that F=m^2/2 – 2
6) a) E(R) for e=2 = 1/3 (10-w) + 2/3 (30-w) = 70/3 – w
E(R) for e=1 = 2/3 (10-w) + 1/3 (30-w) = 50/3 – w
b) For the agent to work hard, with observable effort, U(wh,2) > U(wl,1)
wh^(1/2) – (2-1) > wl^(1/2) – (1-1) => wh^(1/2) > wl^(1/2) + 1
The participation constraints are:
U(wh,2)= u0 = 1
wh^(1/2) – (2-1) = 1 => wh=4
U(wl,1)= u0 = 1
Wl^(1/2) – (1-1) = 1 => wl=1
So, for the agent to work hard, wh>4 and wl=1 and, for the agent to make low effort wh=4 and wl>1
The net revenue for the first case will be
70/3 – 4 = 58/3
and for the second case
50/3 – 1 = 47/3
Forcing contract: U(L,e*) = u0
U(L, 2) = 1 => L^(1/2) – 1 = 1 => L = 4
c) Principal can only observe revenue:
Participation constraint E[U(e=2)] > u0
1/3 wl^(1/2) + 2/3 wh^(1/2) – 1 > 1
1/3 wl^(1/2) + 2/3 wh^(1/2) > 2
wl^(1/2) + 2wh^(1/2) > 6
Incentive compatibility constraint:
E[U(e=2)] > E[U(e=1)]
1/3 wl^(1/2) + 2/3 wh^(1/2) – 1 > 2/3 wl^(1/2) + 1/3 wh^(1/2)
wl^(1/2) + 2wh^(1/2) – 3 > 2wl^(1/2) + wh^(1/2)
-wl^(1/2) + wh^(1/2) > 3
d) Principal’s objective function
max E[R] = 1/3 (10 – wl) + 2/3 (30 – wh)
max 70/3 – wl/3 – 2wh/3
max 70/3 – 1/3(wl+2wh)
is equivalent to
min (wl+2wh) which, in the feasible area, gives wl=0 and wh=9
So, EU agent = 1/3 U(0,2) + 2/3 U(9,2) = 1
E[w] = 1/3. 0 + 2/3.9 = 6
On case b, wage = 4 and expected utility is the same. The wage must be higher to compensate for the additional risk the agent is taking.
e) Expected revenue for e=2 if
1/3 (10-0) + 2/3 (30-9) = 10/3 + 42/3 = 52/3
The expected revenue for e=1 was calculated in item b as 47/3, so motivating is worthwhile.